Minimum phase

A minimum phase system referred to in systems theory a linear time invariant system whose system function has only zeros in the stable region of the complex image plane or in general (also for non-linear systems ) whose zero dynamics is stable. The concept of minimum-phase system applies to both continuous-time and discrete-time systems. They have the further property of having the smallest possible group delay for a given amplitude response.

Continuous-time systems

For continuous-time systems whose transfer function is determined as a Laplace transform of the impulse response of the unstable area of ​​the image plane, the right half-plane with a positive real part. A continuous-time minimum-phase system has only poles and zeros on the left side of the complex half-plane. In other words, a system with a rational transfer function G (s):

Minimum phase if and only if it is stable and no zero digits to the right of the imaginary axis:

Discrete-time systems

For discrete-time systems whose transfer function is determined as the z- transform of the impulse response, the unstable region of the image plane is the one outside the unit circle. A discrete-time minimum-phase system has zeros within the unit circle.

Importance

Minimum phase systems are for example important in the field of control engineering. Non- minimum phase systems can always be decomposed into a minimum-phase component and an all-pass filter, which can lead to a better view of the system and to simplify the development of a controller.